1. § 6.2 Adding and Subtracting Rational Expressions

2. Blitzer, Intermediate Algebra, 4e – Slide #22 Adding Rational Expressions Adding Rational Expressions With Common Denominators If are rational expressions, then To add rational expressions with the same denominator, add numerators and place the sum over the common denominator. If possible, simplify the result.

3. Blitzer, Intermediate Algebra, 4e – Slide #23 Adding Rational ExpressionsEXAMPLE Add: SOLUTION Add numerators. Place this sum over the common denominator. Factor. This is the original expression. Combine like terms.

4. Blitzer, Intermediate Algebra, 4e – Slide #24 Adding Rational Expressions Simplify. Factor and simplify by dividing out the common factor, x. CONTINUED

5. Blitzer, Intermediate Algebra, 4e – Slide #25 Subtracting Rational Expressions Subtracting Rational Expressions With Common Denominators If are rational expressions, then To subtract rational expressions with the same denominator, subtract numerators and place the difference over the common denominator. If possible, simplify the result.

6. Blitzer, Intermediate Algebra, 4e – Slide #26 Subtracting Rational ExpressionsEXAMPLE Subtract: SOLUTION Subtract numerators. Place this difference over the common denominator. This is the original expression. Remove the parentheses and distribute.

7. Blitzer, Intermediate Algebra, 4e – Slide #27 Subtracting Rational Expressions Factor. Combine like terms. Factor and simplify by dividing out the common factor, x + 3. CONTINUED Simplify.

8. Blitzer, Intermediate Algebra, 4e – Slide #28 Least Common Denominators Finding the Least Common Denominator (LCD) 1) Factor each denominator completely. 2) List the factors of the first denominator. 3) Add to the list in step 2 any factors of the second denominator that do not appear in the list. 4) Form the product of each different factor from the list in step 3. This product is the least common denominator.

9. Blitzer, Intermediate Algebra, 4e – Slide #29 Least Common DenominatorsEXAMPLE Find the LCD of: SOLUTION 1) Factor each denominator completely. 2) List the factors of the first denominator.

10. Blitzer, Intermediate Algebra, 4e – Slide #30 Least Common Denominators 3) Add any unlisted factors from the second denominator. The second denominator is (2y - 1)(y + 4). One factor of y + 4 is already in our list, but the factor 2y – 1 is not. We add the factor 2y – 1 to our list. 4) The least common denominator is the product of all factors in the final list. Thus, CONTINUED is the least common denominator.

11. Blitzer, Intermediate Algebra, 4e – Slide #31 Add & Subtract Different Denominators Adding and Subtracting Rational Expressions That Have Different Denominators 1) Find the LCD of the rational expressions. 2) Rewrite each rational expression as an equivalent expression whose denominator is the LCD. To do so, multiply the numerator and denominator of each rational expression by any factor(s) needed to convert the denominator into the LCD. 3) Add or subtract numerators, placing the resulting expression over the LCD. 4) If possible, simplify the resulting rational expressions.

12. Blitzer, Intermediate Algebra, 4e – Slide #32 Adding Different DenominatorsEXAMPLE Add: SOLUTION 1) Find the least common denominator. Begin by factoring the denominators. The factors of the first denominator are x + 4 and x – 2. The only factor from the second denominator that is unlisted is x – 1. Thus, the least common denominator is,

13. Blitzer, Intermediate Algebra, 4e – Slide #33 Adding Different Denominators 2) Write equivalent expressions with the LCD as denominators. This is the original expression. CONTINUED Factored denominators. Multiply each numerator and denominator by the extra factor required to form the LCD.

14. Blitzer, Intermediate Algebra, 4e – Slide #34 Adding Different Denominators 3) & 4) Add numerators, putting this sum over the LCD. Simplify, if possible. Add numerators. CONTINUED Perform the multiplications using the distributive property. Combine like terms.

15. Blitzer, Intermediate Algebra, 4e – Slide #35 Adding Different DenominatorsCONTINUED Since the numerator does not factor, there are clearly no common factors betwixt the numerator and the denominator. Therefore, the final solution is,

16. Blitzer, Intermediate Algebra, 4e – Slide #36 Subtracting Different DenominatorsEXAMPLE Subtract: SOLUTION 1) Find the least common denominator. Begin by factoring the denominators. The factors of the first denominator are x - 4 and x – 1. The only factor from the second denominator that is unlisted is x + 1. Thus, the least common denominator is,

17. Blitzer, Intermediate Algebra, 4e – Slide #37 Subtracting Different Denominators 2) Write equivalent expressions with the LCD as denominators. This is the original expression. CONTINUED Factored denominators. Multiply each numerator and denominator by the extra factor required to form the LCD.

18. Blitzer, Intermediate Algebra, 4e – Slide #38 Subtracting Different Denominators 3) & 4) Add numerators, putting this sum over the LCD. Simplify, if possible. Subtract numerators. CONTINUED Perform the multiplications using the distributive property and FOIL. Remove parentheses.

19. Blitzer, Intermediate Algebra, 4e – Slide #39 Subtracting Different Denominators Combine like terms in the numerator. CONTINUED Since the numerator does not factor, there are clearly no common factors betwixt the numerator and the denominator. Therefore, the final solution is,

20. Blitzer, Intermediate Algebra, 4e – Slide #40 Add & Subtract Different DenominatorsEXAMPLE Perform the indicated operations: SOLUTION 1) Find the least common denominator. Begin by factoring the denominators. The factors of the first denominator are 1 and x – 3. The only factor from the second denominator that is unlisted is x + 1. We have already listed all factors from the third denominator. Thus, the least common denominator is,

21. Blitzer, Intermediate Algebra, 4e – Slide #41 Add & Subtract Different DenominatorsCONTINUED 2) Write equivalent expressions with the LCD as denominator. This is the original expression. Factor the second denominator. Multiply each numerator and denominator by the extra factor required to form the LCD.

22. Blitzer, Intermediate Algebra, 4e – Slide #42 Add & Subtract Different DenominatorsCONTINUED 3) & 4) Add and subtract numerators, putting this result over the LCD. Simplify if possible. Add and subtract numerators. Perform the multiplications using the distributive property. Combine like terms in the numerator.

23. Blitzer, Intermediate Algebra, 4e – Slide #43 Add & Subtract Different DenominatorsCONTINUED Since the numerator does not factor, there are clearly no common factors betwixt the numerator and the denominator. Therefore, the final solution is,

24. Blitzer, Intermediate Algebra, 4e – Slide #44 Add & Subtract Opposite DenominatorsEXAMPLE Add: SOLUTION This is the original expression. Factor the first denominator. Multiply the numerator and the denominator of the second rational expression by -1. Multiply by -1.

25. Blitzer, Intermediate Algebra, 4e – Slide #45 Add & Subtract Opposite Denominators Rewrite –y + x as x – y. Notice the LCD is (x + y)(x – y). Multiply the second numerator and denominator by the extra factor required to form the LCD. Perform the multiplications using the distributive property. CONTINUED

26. Blitzer, Intermediate Algebra, 4e – Slide #46 Add & Subtract Opposite Denominators Add and subtract numerators. Remove parentheses and distribute. Combine like terms in the numerator. Since the numerator does not factor, there are clearly no common factors betwixt the numerator and the denominator. Therefore, the final solution is, CONTINUED